A Free Boundary Problem Related to Condensed Two-Phase Combustion. Part I. Semigroup

We study the dynamical aspects of a general one space dimension two-phase nonlinear free boundary problem which is related to condensed two-phase combustion. Among those, we give particular attention to the dynamics near traveling wave solutions. By reducing the free boundary problem to a coupled system consisting of two reaction diffusion equations and one ordinary differential equation, we prove that the linearized system of a traveling wave solution defines an analytic semigroup in a Banach space and the essential spectrum of the corresponding infinitesimal generator can be controlled within the left halfplane in appropriate weighted norm spaces. Using the analytic semigroup theory, we also prove that the nonlinear system defines a local semigroup in some fractional power space